Reference request: overdetermined polynomial systems are almost surely inconsistent If one has $m$ multi-variate polynomial equations with $n$ variables, and $m>n$ one obtains an overdetermined system.
According to wikipedia here "Most but not all overdetermined systems, when constructed with random coefficients, are inconsistent." Is this equivalent to say that among the space of coefficients for these polynomials, the set of overdetermined systems with solution is of measure zero? Or equivalently, that when choosing random coefficients, overdetermined systems are inconsistent almost surely?
If this is the case. Do you know a canonical reference (book/article) for this fact for me to cite? (The references in the wikipedia page didn't helped much).
Why I started a bounty? while the answer by @orangeskid (and the reference he provided) is awesome in explaining why this fact is true, I am limited in space in the manuscript I'm writing and I cannot add much more detail to it.
Note that the manuscript I'm writing lies in an area completely different from algebraic geometry and resultant theory, so introducing the resultant there might be confusing for the reader (And my advisor did not liked it). That is why I would like to keep as less details as I can regarding why the fact is true, while being sure that the sentence is true and correctly referenced.
Ideally, I would like to know if there is a reference one could use for a sentence like

Recall that the set of over-determined polynomial systems of equations are inconsistent for most coefficients used in them (in the sense of the Lebesgue measure for the coefficients space). [INSERT CITATION HERE]

Or something like that. Perhaps being still an amateur, I wasn't able to see if something in the reference @orangeskid provided can be cited directly. Maybe there is no other way but to add more detail (which I would like to avoid) as in the answer from @orangeskid but I would like to hear your suggestions!
 A: This involves the resultant of $N$ homogenous forms in $N$ variables.
For consider $n+1$ polynomials in $n$ variables $t_1$, $\ldots$, $t_n$. Homogenize and get $N$ forms in $N$ variables $x_0$, $\ldots$, $x_n$ ( $N=n+1$).
If the $n+1$ polynomials have a common solution then the $n+1$ homogenous forms have a  solution ( not at infinity). It follows that their resultant is $0$  (the converse : zero resultant implies there exists a solution, but it can be "at infinity"). Nevertheless, we see that if the polynomials have a common solution, then some non-zero polynomial in their coefficients is $0$.  But the set of zeroes of the resultant is small ( for instance, it has measure $0$).
A concrete example: consider a system of $n+1$ linear equations with $n$ unknowns $A t = b$. If it has a solution then the extended $(n+1)\times (n+1)$ matrix $(A,b)$ has $0$ determinant.
$\bf{Added:}$
Perhaps it is worth adding the proof of the following result.
If $P\in \mathbb{R}[X_1, \ldots, X_n]$ is a non-zero polynomial then the zero-set of $P$
$$Z(P) \colon = \{t=(t_1,\ldots, t_n)\in \mathbb{R}^n\ | \ P(t) = 0\}$$
has Lebesgue measure $0$.
Decompose $P$ into homogenous components:
$P=P_0+ \cdots + P_d$
$P_d$ homogenous of degree $d = \deg P$.  Assume that $P_d$ contains the monomial $x_n^d$ with a non-zero coefficient. Otherwise, apply a linear transformation of coordinates so this becomes the case. So we may assume
$$P(x) = x_n^d + Q_1(x_1,\ldots, x_{n-1}) x_n^{d-1} + \cdots + Q_d(x_1,\ldots, x_{n-1})$$
Now, the intersection of $Z(P)$ with any line vertical line $(x_1, \ldots, x_{n-1}) = (a_1, \ldots, a_{n-1})$ has at most $d$ points, and so measure $0$ in that line. Now apply Fubini.
